Problem: Estimating $e^{1.3}$ using a Taylor polynomial about $x=1$, what is the least degree of the polynomial that assures an error smaller than $0.001$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $2$ (Choice B) B $3$ (Choice C) C $4$ (Choice D) D $5$
We will use the Lagrange error bound. Let's assume the polynomial's degree is $n$. The $(n+1)^{\text{th}}$ derivative of $e^x$ is $e^x$. On the interval between $x=1$ and $x=1.3$, the greatest value of the derivative is $e^{1.3}\approx3.7$. The Lagrange bound for the error assures that $|R_n(1.3)|\leq \dfrac{3.7}{(n+1)!}(1.3-1)^{n+1}$. Solving $\dfrac{3.7}{(n+1)!}0.3^{n+1}<0.001$ using trial and error, we find that $n\geq4$. In conclusion, the least degree of the polynomial that assures our error bound is $4$.